Mistaking a map for the territory
A while back, I got a copy of Circuit Analysis by Robbins and Miller, to review material that I'd forgot, and to fill-in lacunæ here and there. On the whole, I think that it's a pretty good book, though somewhat slow-moving for my tastes.
But to-day I hit a passage that bugs me:
Although we use phasors to represent sinusoidal waveforms, it should be noted that sine waves and phasors are not the same thing. Sinusoidal voltages and currents are real — they are actual quantities that you measure with meters and whose waveforms you see on oscilloscopes. Phasors, on the other hand, are mathematical abstractions that we use to help visualize relationships and solve problems.
Okay, now, for those of you unfamiliar with phasors, these are two-dimensional vectors or complex numbers whose magnitude corresponds to the amplitude of a sinus wave, and whose direction corresponds to the phase of the wave. In the case of a wave with an amplitude of 1 unit, a phasor would be a radial line on a unit circle, bearing the familiar relationship to a sine wave.
There is an isomorphism between the set of phasors and any representation of sinus waves. That is to say that for every representation of sinus waves and operations thereüpon, one can find equivalent phasors and operations thereüpon, and vice versa, such that a one-to-one correspondence between operands and results is maintained. From the perspective of mathematics, a phasor just is a representation of a sinus wave.
This last point does not contradict what Robbins and Miller have said, but now consider how and why we see wave-forms on an oscilloscope. The most familiar graphical representation of sinus waves looks very much like some of the waves that we observe in water, but electricity isn't water. We cannot look at an ordinary circuit and see its voltage or current; instead, we use devices whose visible behaviour changes to represent voltage or current. These devices might represent that behaviour in various ways; the ways in which they do are determined by various cost considerations (including cultural expectations).
An oscilloscope is designed to present a particular sort of graphical representation of wave-forms. It could instead be designed to present a different sort of graphical representation. If it only had to represent sinus waves, then it could do this with stable phasors. And if it had to represent non-sinus waves, then it could perhaps do this with time-varying phasors (giving the viewer an animated Fourier analysis), though I don't know that this would be as helpful to us.
The representation of the oscilloscope is a map. As Korzybski noted, the map is not the territory. The phasor is not the voltage or the current, but neither is the representation on the oscilloscope; neither is more or less real than is the other.